Here \(\nabla \hspace{-1.38885pt}_{s:t,i}\)  denotes the \(ith\)  coordinate of \(\sum _{\tau =s}^t \nabla \hspace{-1.38885pt}_{\tau }\) .

The proof is almost identical to that of the previous corollary, observing that the regret \(R_0(I)\)  is \(\tilde{O}(D_{\infty } \sum _{i=1}^d \Vert \nabla \hspace{-1.38885pt}_{s:t,i}\Vert _2 )\)  due to [1]}, and the regret \(R_1(I)\)  remains \(\tilde{O}(D \sqrt{ \min _{H\in {\mathcal {H}}} \sum _{\tau =s}^t \nabla \hspace{-1.38885pt}_{\tau }^{\top } H^{-1} \nabla \hspace{-1.38885pt}_{\tau }})\) , which is upper bounded by \(\tilde{O}(D_{\infty } \sum _{i=1}^d \Vert \nabla \hspace{-1.38885pt}_{s:t,i}\Vert _2 )\) .
